Higher Technical Institute, Nicosia, Cyprus. Received October 1982; revised version received February 1983. Abstract. A generalized geometric distribution is ...
Statistics & Probability Letters 1 (1983) 171-175 North-Holland
June 1983
A Generalized Geometric Distribution and some of its Properties * Andreas N. Philippou and Costas Georghiou Department of Mathematics, University of Patras, Patras, Greece
George N. Philippou Higher Technical Institute, Nicosia, Cyprus
Received October 1982; revised version received February 1983
Abstract. A generalized geometric distribution is introduced and briefly studied. First it is noted that it is a proper probability distribution. Then its probability generating function, mean and variance are derived. The probability distribution of the sum Y, of r independent random variables, distributed as generalized geometric, is derived. Finally, sufficient conditions are presented under which we can derive the limiting distribution of Y, - kr as r ~ o0. Keywords. Generalized geometric distribution, probability generating function, mean, variance, generalized negative binomial distribution, generalized Poisson distribution.
1. Introduction and summary In this p a p e r k a n d r are fixed integers greater than or equal to 1, unless otherwise stated, x, (1 ~< i ~< k ) a n d x are nonnegative integers as specified, a n d p is a real n u m b e r in the interval (0, 1). M o t i v a t e d by the recent w o r k of P h i l i p p o u and M u w a f i (1982), in Section 2 we i n t r o d u c e the geometric d i s t r i b u t i o n of o r d e r k with p a r a m e t e r p (to be d e n o t e d b y G k ( ' ; p ) ) . First it is o b s e r v e d that G l ( . ; p ) is the usual geometric distribution, which shows that G k ( - ; p ) is a generalized geometric distribution. N e x t it is n o t e d that G k ( . ; p ) is a p r o p e r p r o b a b i l i t y distribution, and its p r o b a b i l i t y generating function, m e a n and variance are derived. In Section 3 we derive the p r o b a b i l i t y d i s t r i b u t i o n of the sum Y, of r i n d e p e n d e n t r a n d o m variables d i s t r i b u t e d as g e o m e t r i c of o r d e r k. This d i s t r i b u t i o n will be referred to as the negative b i n o m i a l d i s t r i b u t i o n or o r d e r k with p a r a m e t e r vector (r, p ) . Finally, u n d e r certain regularity c o n d i t i o n s we o b t a i n the limiting d i s t r i b u t i o n of Y , - k r at r--* oo. This d i s t r i b u t i o n will be referred to as the Poisson d i s t r i b u t i o n of o r d e r k. In ending this section we note that the present w o r k generalizes w e l l - k n o w n results on the usual geometric distribution.
* A preliminary version of this paper was presented at the 7th Conference on Probability Theory held in Brasov, Romania, from August 29 to September 4, 1982. 0167-7152/83/$03.00 © 1983 Elsevier Science Publishers B.V. (North-Holland)
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2. The geometric distribution of order k In this section, the geometric distribution of order k is introduced and some of its properties are examined. Definition 2.1. A r a n d o m variable X is said to have the geometric distribution of order k with p a r a m e t e r p, to be denoted by G,(x; p), if
xl .....
xk
Xl'''''Xk
pX
,
x>~k,
where the s u m m a t i o n is over all nonnegative integers x t , . . . , x k such that x~ + 2x 2 + • • • + kx k = x - k, and q = 1 - p . If we denote the geometric distribution by G ( - ; p), it follows from Definition 2.1 that GI(X;p ) = G(x; p). Therefore G , ( . ; p ) is a generalized geometric distribution. It can be shown that Gk(" ; p ) is a proper probability distribution, which implies that its probability generating function exists for Isl ~ 1. In fact, we have the following result. L e m m a 2.2. Let X be a random variable distributed as G,(x; p). Then its probability generating function, denoted by yk(s), is given by
Yk(S)=pks*(1--ps)/[l--s+qpks*+l],
Isl~ 1.
Proof. We have Es
P(X=
sx+ r'(x=x+k)
)=
x~k
x~O
(by Definition 2.1) x~O
x I.....
xl+2x2+
=PkSk ~ n~0
x~ ~
...
X l ' " " " ' xk
--
+kx~=x
-(nl+n,,.'''.,n k+nk)( p s ) n~+ 2n2+ ... +k,,~ ( q ) " ' + + " '
~" n I..... nk~ r/14- - - .
\
(
k
4-r/k~t/
by s e t t i n g x s = n~ (1 ~< i~< k ) and x = n + Y'. ( i -
) l)n,
i=l
= pksk
( ps + p2s2 + "" + pks* ) n~0
=pksk
1
1 - - q ( p s +p2s2 + ... +pksk ) P
=pks*(l -ps)/[l
]
(by the multinomial theorem) (since Isl ~ 1)
- s + qp~s k-'].
We can now employ L e m m a 2.2 to obtain the mean and variance of the geometric distribution of order k by straightforward differentiation of y,(s). 172
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Proposition 2.3. Let X be a random variable distributed as G k( X; p ). Then E(X)=(1--pk)/qpk
and
o2(X)=[1-(2k+l)qp
k-p2k+l]/q2p2k.
Corollary 2.4. When k = 1, the mean and variance of the geometric distribution are obtained. Remark 2.5. Let X~. . . . . Xn be independent identically distributed random variables as Gk(X; p ) and set n )~= ( I / n ) Y],j= IXj. Proposition 2.3 may be employed to obtain the moment estimator p of p. In fact,/~ is the unique admissible root of the equation ( 1 - p k ) / [ ( 1 _ p ) p k ] = ~. Thus we have, for k = 2 , p = [1 + (1 + 4.,~)z/2]/(2.~), which is consistent f o r p . We end this section by noting the following proposition which describes how the geometric distribution of order k may arise. Proposition 2.6. Let N k be a random variable denoting the number of trials until the occurrence of the k-th consecutive success in independent trials with success probability p. Then N k is distributed as G k( n; p ).
ProoL For k = 1, it is well known that N 1 is G(n; p). But G(n; p ) = G~(n; p ) by Definition 2.1, so that N l is Gl(n; p). For k >/2, N k is Gk(n; p ) because of Theorem 3.1 of Philippou and Muwafi (1982) and our Definition 2.1. Remark 2.7. Shane (1973) was able to derive the probability generating function of Nk, and hence its mean, by developing a recursion formula for P ( N k = n), n >/k, in terms of his Polynacci polynomials of order k in p. 3. The negative binomial and Poisson distribution of order k In this section we derive the probability distribution of the sum Yr of r independent random variables distributed as geometric of order k. This probability distribution is a generalized negative binomial distribution. We also obtain the limiting distribution of Y r - k r as r--* ~ , which is shown to be a generalized Poisson distribution. Theorem 3.1. Let X I . . . . . X r be independent random variables distributed as G k ( X; p ) and set Yr -= ~,= IX,. Then v, ..... y~
Yl . . . . . Yk , r -
1
'I
y >~kr,
PY
where the summation is over all nonnegative integers y/, . . . ,Yk such that
y~+2y 2+.--
+ky k=y-kr.
Proof. We observe that OO
E
sYP(Yr=Y)
y =kr r
= E(svr) = I--I E ( s x ' )
(by the definition of Yr and the independence of X, . . . . . Xr)
i~l
pksk(1 - - p s ) 1 - S + C l p k s k+~
pkrskr r
(by Lemma 2.2)
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E ( -,‘)[-;(ps+p2s2+ .. . +p’sk]m
=pkrsk’
m-0
=pkTsXrm~o(m+~-l)(~)m[pS+p%~+ .*f+p5qm _krskr
f
(“+;-
I)(
$
c
m-0
(m;:
m,,...,mk3 m,+
“’
~~k)(~~,-,+2m*+...+km~
theorem)
E rn,.....tnk3 ??I,+
,..
(
+mk=m
03
cpkrsk’
c
C y-kr
yc+2y2+
( ps)y-kr
Y,,...>Yk3
...
+kyk=y-kr
Y, + * * * y,+2yz+
i (ii=l
